A 4.0 kg bowling ball rolling at a velocity of 11m/s down the bowling lane makes a head on collision with a 2.5 kg bowling pin initially at rest. After the collision, the pink has a velocity of 13.54m/s down the bowling lane. What is the final velocity of the bowling ball?

What type of Collision is this? Confirm mathematically!

momentum:

4*11+0= 2.5*13.54 + 4V
solve for V.

To confirm mathmetically, set initial KE equal to final KE, and see if they equal.

To find the final velocity of the bowling ball after the collision, we can use the law of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is calculated by multiplying its mass by its velocity. Therefore, the initial momentum of the bowling ball is given by:
P_initial = m_bowling ball * v_initial
where:
m_bowling ball is the mass of the bowling ball (4.0 kg)
v_initial is the initial velocity of the bowling ball (11 m/s)

The momentum of the bowling pin is initially zero since it is at rest:
P_initial = m_pin * v_pin
where:
m_pin is the mass of the bowling pin (2.5 kg)
v_pin is the initial velocity of the bowling pin (0 m/s)

After the collision, the final momentum of the bowling ball is given by:
P_final = m_bowling ball * v_final
where:
v_final is the final velocity of the bowling ball (unknown)

The final momentum of the bowling pin is calculated as:
P_final = m_pin * v_pin
where:
v_pin is the final velocity of the bowling pin (13.54 m/s)

According to the law of conservation of momentum:
P_initial = P_final
m_bowling ball * v_initial = m_bowling ball * v_final

Plugging in the known values:
4.0 kg * 11 m/s = 4.0 kg * v_final

Now we can solve for v_final:
v_final = (4.0 kg * 11 m/s) / 4.0 kg
v_final = 44 m/s

Therefore, the final velocity of the bowling ball is 44 m/s.

Now let's determine the type of collision mathematically. In an elastic collision, both kinetic energy and momentum are conserved. In an inelastic collision, momentum is conserved, but kinetic energy is not conserved.

To check for conservation of kinetic energy, we can calculate the initial and final kinetic energy of the system.

The initial kinetic energy is given by:
KE_initial = (1/2) * m_bowling ball * v_initial^2 + (1/2) * m_pin * v_pin^2

The final kinetic energy is given by:
KE_final = (1/2) * m_bowling ball * v_final^2 + (1/2) * m_pin * v_pin^2

If KE_initial = KE_final, then it is an elastic collision. If KE_initial > KE_final, then it is an inelastic collision.

By plugging in the given values and calculating the kinetic energy before and after the collision, we can conclude the type of collision.